The prelimit generator comparison approach of Stein's method

TL;DR

This paper introduces a prelimit generator comparison approach in Stein's method, focusing on finite differences of the Markov chain's Poisson equation to analyze distribution gaps more easily.

Contribution

It proposes starting with the Markov chain's Poisson equation instead of the diffusion's, simplifying the derivation of bounds in Stein's method.

Findings

The prelimit approach simplifies obtaining Stein factor bounds.

Application to M/M/1 model demonstrates the method's effectiveness.

Finite difference bounds are easier to derive than derivative bounds.

Abstract

This paper uses the generator comparison approach of Stein's method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. The "standard" generator comparison approach starts with the Poisson equation for the diffusion, and the main technical difficulty is to obtain bounds on the derivatives of the solution to the Poisson equation, also known as Stein factor bounds. In this paper we propose starting with the Poisson equation of the Markov chain; we term this the prelimit approach. Although one still needs Stein factor bounds, they now correspond to finite differences of the Markov chain Poisson equation solution rather than the derivatives of the solution to the diffusion Poisson equation. In certain cases, the former are easier to obtain. We use the $M/M/1$ model as a simple working example to illustrate our approach.